Monte Carlo simulations of imaginary time Liouvillian dynamics in the mixed field Ising model.*

Under generic Hamiltonian dynamics, governing the evolution of interacting many-body systems, local operators are expected to rapidly evolve in time, resulting in complex structures of operator strings. Understanding the properties of this operator growth process has attracted significant research interest, especially in relation to thermalization in quantum systems. Here, we present a Monte Carlo scheme which enables probing operator growth by sampling the imaginary time Liouvillian dynamics. Compared to more conventional quantum Monte Carlo schemes, our configuration space comprises local operators occupying a D+1 dimensional lattice instead of wavefunctions. Transitions between operator strings are captured by the action of the Liouvillian. Crucially, our algorithm is free of the numerical sign problem for the mixed field quantum Ising model at infinite temperature, allowing for numerically exact calculations. Our findings support the recently proposed "Operator Growth Hypothesis", predicting a high-frequency exponential tail for generic local spectral functions. In particular, we resolve subtle logarithmic corrections in 1D and a crossover in the high-frequency decay rates in 2D. Extension of our approach to other models and observables will be discussed.